Energy due to position of a particle is given by, `U=(alpha sqrty)/(y+beta)`, where `alpha` and `beta` are constants, y is distance. The dimensions of `(alpha xx beta)` are

Options

A. `[M^(0)LT^(0)]`
B. `[M^(1//2)L^(3//2)T^(-2)]`
C. `[M^(0)L^(-7//2)T^(0)]`
D. `[ML^(7//2)T^(-2)]`

Answer

To find the dimensions of the product \( \alpha \times \beta \) given the energy expression \( U = \frac{\alpha \sqrt{y}}{y + \beta} \), we can follow these steps:

### Step 1: Identify the dimensions of the variables
- The variable \( U \) represents energy, which has the dimension of \( [U] = [M][L^2][T^{-2}] \).
- The variable \( y \) represents distance, which has the dimension of \( [y] = [L] \).

### Step 2: Analyze the denominator \( y + \beta \)
- The term \( y + \beta \) must have the same dimensions. Since \( y \) has the dimension of length \( [L] \), \( \beta \) must also have the dimension of length. Therefore, we can conclude:
\[ [\beta] = [L] \]

### Step 3: Rearranging the equation for \( \alpha \)
- The equation can be rearranged to express \( \alpha \):
\[ U = \frac{\alpha \sqrt{y}}{y + \beta} \]
Since \( y + \beta \) has the dimension of \( [L] \), we can rewrite the equation as:
\[ U = \frac{\alpha \sqrt{y}}{[L]} \]
This implies:
\[ [U] = [\alpha] \cdot [\sqrt{y}] \cdot [L^{-1}] \]

### Step 4: Substitute the dimensions
- Substituting the known dimensions:
\[ [U] = [\alpha] \cdot [L^{1/2}] \cdot [L^{-1}] \]
This simplifies to:
\[ [U] = [\alpha] \cdot [L^{-1/2}] \]

### Step 5: Solve for \( [\alpha] \)
- Rearranging gives:
\[ [\alpha] = [U] \cdot [L^{1/2}] \]
Substituting the dimension of energy:
\[ [\alpha] = [M][L^2][T^{-2}] \cdot [L^{1/2}] = [M][L^{2 + 1/2}][T^{-2}] = [M][L^{5/2}][T^{-2}] \]

### Step 6: Find the dimensions of \( \alpha \times \beta \)
- Now we can find the dimensions of \( \alpha \times \beta \):
\[ [\alpha \times \beta] = [\alpha] \cdot [\beta] = ([M][L^{5/2}][T^{-2}]) \cdot ([L]) \]
This results in:
\[ [\alpha \times \beta] = [M][L^{5/2 + 1}][T^{-2}] = [M][L^{7/2}][T^{-2}] \]

### Final Answer
The dimensions of \( \alpha \times \beta \) are:
\[ [M][L^{7/2}][T^{-2}] \]